Definition 1 (Asymptotic continuity [1]}, [2]}) 
Let \(f: {\cal D}(*) \rightarrow \mathbb {R}_{+}.\) 
We say that \(f\)  is asymptotically continuous if there exists a constant \(K\)  such that, for any \(d\)  and any
\(\mu , \nu \in {\cal D}(d)\) ,
\(|f(\mu ) - f(\nu )| \le K\, \Vert  \mu - \nu \Vert _1 \log d + \eta (\Vert  \mu - \nu \Vert _1)\) 
